Tourish 的一个根式不等式
好久没更新博客了,这段时间比较忙。以后争取有空多贴点好的问题,多谢大家光临我的博客。
Problem
If a,b,c are positive real numbers such that a+b+c=3,then
√a3+3b+√b3+3c+√c3+3a≥6
Proof. By the Cauchy-Schwarz inequality,we have
(a3+3b)(a+3b)≥(a2+3b)2
Thus,it suffices to show that
∑a2+3b√a+3b≥6
By Holder inequality,we have
(∑a2+3b√a+3b)2[∑(a2+3b)(a+3b)]≥[∑(a2+3b)]3=(∑a2+9)3
Therefore,it is enough to show that
(∑a2+9)3≥36∑(a2+3b)(a+3b)
Let p=a+b+c=3 and q=ab+bc+ca,q≤3.We have
∑a2+9=p2−2q+9=2(9−q)
∑(a2+3b)(a+3b)=∑a3+3∑a2b+9∑a2+3∑ab=(p3−3pq+3abc)+3∑a2b+9(p2−2q)+3q=108−24q+3(abc+∑a2b)
Since
a2b+b2c+c2a+abc≤427(a+b+c)3
We get
∑(a2+3b)(a+3b)≤24(5−q)
Thus.it suffices to show that
(9−a)3≥108(5−a)
Which is equivalent to the obvious inequality
(3−q)2(21−q)≥0
The equality holds for a=b=c=1.